Incenter | 合法醫療器材資訊網

Theincenterofatriangleisthecenterofitsinscribedcircle.Ithasseveralimportantpropertiesandrelationswithotherpartsofthetriangle,includingits ...

The simplest proof is a consequence of the trigonometric version of Cevas theorem[1], which states that (AD, BE, CF) concur if and only if

[frac{sinangle BAD}{sinangle ABE} cdot frac{sin angle CBE}{sin angle BCF} cdot frac{sinangle ACF}{sin angle CAD} = 1.]

In this case, (D,E,F) are the feet of the angle bisectors, so (angle BAD=angle CAD), (angle ABE=angle CBE), and (angle ACF=angle BCF). As a result,

[frac{sinangle BAD}{sinangle CAD} cdot frac{sinangle ABE}{sinangle CBE} cdot frac{sinangle ACF}{sinangle BCF} = 1 cdot 1 cdot 1 = 1]

and rearranging the left hand side gives

[frac{sinangle BAD}{sinangle ABE} cdot frac{sin angle CBE}{sin angle BCF} cdot frac{sinangle ACF}{sin angle CAD} = 1.]

Therefore, the three angle bisectors intersect at a single point,...